Chris Schommer-Pries has classified 2d extended TQFTs (topological quantum field theories) in his PhD thesis. The result is a (not necessarily abelian) separable symmetric Frobenius algebra (possibly internal to some symmetric monoidal category).
What if we add defects into the game? I couldn't find a hard classification result for 2d extended TQFTs with defects. Is there one? Or at least a folk theorem? Does the classification of extended TQFTs without defects somehow follow as a special case?
EDIT: If there are hints on how the classification for non-extended 2d TQFTs (2-1-TQFTs, so to speak) with defects goes, that would be relevant too.